Weighted identity for Howard's semicircle theorem ($\int (U-c{)}^2(|F^{\prime }{|}^2+k^2|F{|}^2)dz=0$)

For a nonreal regular Rayleigh equation for inviscid shear flow mode vanishing at the walls, put $F=\hat{w}/(U-c)$. The equation becomes $[(U-c{)}^2{F}^{\prime }{]}^{\prime }-k^2(U-c{)}^{2}F=0$. Multiplication by $\overline{F}$ and integration by parts gives the displayed identity. With $Q=|F^{\prime }{|}^2+k^2|F{|}^2$, its imaginary and real parts give $\int UQ=c_r\int Q$ and $\int U^{2}Q=|c{|}^2\int Q$. Since $(U-U_{\min })(U_{\max }-U)\ge 0$, these imply Howard's semicircle theorem.